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Properties

Label 1-111-111.35-r0-0-0
Degree $1$
Conductor $111$
Sign $0.995 - 0.0953i$
Analytic cond. $0.515481$
Root an. cond. $0.515481$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.642 − 0.766i)5-s + (0.766 − 0.642i)7-s + (0.866 + 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.342 − 0.939i)13-s + (0.866 − 0.5i)14-s + (0.766 + 0.642i)16-s + (0.342 + 0.939i)17-s + (−0.984 + 0.173i)19-s + (−0.342 − 0.939i)20-s + (−0.642 + 0.766i)22-s + (−0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.642 − 0.766i)5-s + (0.766 − 0.642i)7-s + (0.866 + 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.342 − 0.939i)13-s + (0.866 − 0.5i)14-s + (0.766 + 0.642i)16-s + (0.342 + 0.939i)17-s + (−0.984 + 0.173i)19-s + (−0.342 − 0.939i)20-s + (−0.642 + 0.766i)22-s + (−0.866 + 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.995 - 0.0953i$
Analytic conductor: \(0.515481\)
Root analytic conductor: \(0.515481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (0:\ ),\ 0.995 - 0.0953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.743465261 - 0.08328697710i\)
\(L(\frac12)\) \(\approx\) \(1.743465261 - 0.08328697710i\)
\(L(1)\) \(\approx\) \(1.677937917 + 0.02118761036i\)
\(L(1)\) \(\approx\) \(1.677937917 + 0.02118761036i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
5 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.342 - 0.939i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (0.642 - 0.766i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 - T \)
79 \( 1 + (0.642 + 0.766i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.779467293217533210110174582425, −28.61224762916563692145429502018, −27.530910766336763648456498225986, −26.352446585175161291676776250475, −25.18734144013051989962913346866, −23.960508876468939673287758643175, −23.46738512814173341821839122407, −22.13110996756868548042653020256, −21.46622137891676902072682240136, −20.402804244456466690233170988918, −19.03024880787325871658380589675, −18.40329920691197119781091337, −16.468561455944228004163282342411, −15.53895714122940737783670985934, −14.536019979554622032335866634108, −13.75380048961261354539841809462, −12.18422311849354539477985263959, −11.39269891246136272894840467492, −10.54460699123332370458188549645, −8.560547487780215220956472205211, −7.23780852433983676871485113361, −6.02705871397096934475118494781, −4.71459664534313374944210896309, −3.39994295291918906938114546743, −2.13531337102251474395769251222, 1.78447009789676705486081335680, 3.765715894379716672160230030687, 4.6312383228602887678012047127, 5.83704331563918706530893900935, 7.58929513236281465376139240118, 8.15054450776068222940816587461, 10.29810811564453882458247207372, 11.40787845023094087550877760176, 12.58206843617100581041669860018, 13.29798336620951836881986862869, 14.78475146369065951581057837585, 15.4597396958911640312968452082, 16.71753439097496045008453546303, 17.60108379005367992985244133728, 19.45748949826558885101302874890, 20.532719542704272226185890798574, 20.926650109536433889505996314854, 22.44936821716673734265401544036, 23.565795416893157210295434909021, 23.89238556401654785669199104929, 25.152318521535251081483008162, 26.1356090806147460704844812636, 27.62699110756913475736556787109, 28.34320356344548778546472284279, 29.863365725677719913067023277054

Graph of the $Z$-function along the critical line